Blaschke products and the rank of backward shift invariant subspaces over the bidisk (Q639526)

Let \(H^2(\mathbb{D}^2)\) be the Hardy space over the bidisk. With \(T_z\) and \(T_w\) we denote the multiplication operators by the variables \(z\) and \(w\), respectively. A closed subspace \(M\) of \(H^2(\mathbb{D}^2)\) is said to be invariant if it is invariant for operators \(T_z\) and \(T_w\). For a nonempty subset \(E\) of an invariant subspace \(M\), let \([E]_T\) be the smallest invariant subspace of \(H^2(\mathbb{D}^2)\) which contains \(E\). If \([E]_T=M\), then \(E\) is said to be a generating set of \(M\). Let \(\# E\) be the number of elements in \(E\). The rank of \(M\) is defined as \[ \text{rank}_T M:=\inf\limits_E \# E, \] where the infimum on the right hand side is taken over all generating sets of \(M\). The rank \(\text{rank}_{T^*} M^{\perp}\) is defined similarly. In this paper \(\text{rank}_{T^*} M^{\perp}\) is determined for different invariant subspaces of \(H^2(\mathbb{D}^2)\) which are defined by sequences of Blaschke products.